Technique for reconstructing a surface shape for measuring coordinates

Technique for reconstructing a surface shape for measuring coordinates

V. Kh. Sa nov,a兲Zh. I. Kharizanova, and E. V. Sto kova

Central Laboratory of Optical Recording and Data Processing, Sofia, Bulgaria

Kh. M. Ozaktas and L. Onural

Bilkent University, Ankara, Turkey

共Submitted December 27, 2005兲

Opticheski Zhurnal 73, 49–54 共July 2006兲

This paper describes a method of projecting interference fringes as one of the most accessible techniques for measuring the coordinates of objects and scenes that can be used when solving inverse problems in dynamic holographic display, where the coordinates need to be measured in order to compute diffraction structures when reconstructing three-dimensional images. A com-parative analysis is presented of the experimental results obtained with successive projections of interference patterns with two different periods, using a Mach–Zehnder interferometer in coher-ent light and a micromirror projector with digital generation of fringes in white light. The use of the method is limited by the size of the objects and scenes. The possibilities of using more re-fined methods, including the holographic approach to phase reconstruction, are discussed. © 2006 Optical Society of America.

PHASE RECONSTRUCTION FOR MEASURING ABSOLUTE COORDINATES

Accurate information on the coordinates of objects and three-dimensional scenes is needed in order to implement a dynamic holographic display. The main requirement in this case is to make measurements of a random distribution of objects in a scene in real time. Optical profilometry methods offer the most promising solution of this problem. Some of the optical systems that exist and are for sale for measuring the coordinates of actual objects have been developed on the basis of laser scanning and triangulation.1,2Because the sur-face is scanned in a single direction “point to point” with a limited information-readout rate, the state of the surround-ings 共vibrations, air turbulence, etc.兲 has a strong effect on the measurement accuracy, especially in large-scale scenes under actual conditions. Special attention should be paid to the wide class of phase-reconstruction methods based on moiré and projection interferometry.3,4 The main advantage of these methods is the simplicity, high accuracy, and rate of the measurements.

Information on the surface of an object can be obtained by reconstructing the phase of an object wave front and by recording the intensities in spatial and frequency domains. Multiplicity in spatial frequencies in profilometry based on the Fourier transformation, in combination with a compli-cated phase-reconstruction algorithm, provides measurement in real time by a single recording, and this is very essential when measuring moving objects isolated in space.5A disad-vantage of this method is the comparatively low measure-ment accuracy, which limits its wide practical use. From this viewpoint, the technique of projecting interference fringes with the introduction of a phase shift4 is more suitable for measuring and reconstructing three-dimensional scenes. This simplified approach is widely used in optical metrology for surface measurement and nondestructive testing.6–9 The fringes can be projected with coherent light, using, for

ex-ample, a Michelson interferometer, or in incoherent light by means of two-coordinate spatial light modulators, such as liquid-crystal or micromirror projectors. The recording of fringes deformed by an object’s surface for a given phase shift␾makes it possible to reconstruct the phase⌬␸ modu-lated by the object, using the interferogram:

I共x,y兲 = I¯共x,y兲兵1 +␩共x,y兲cos关⌬␸共x,y兲 +␾兴其, 共1兲 where I共x,y兲 is the intensity at point 共x,y兲, I¯共x,y兲 is the averaged intensity, and ␩ is the contrast of the interference fringes. At least three independent measurements are needed for this purpose, to determine the three unknown parameters in Eq. 共1兲. Various phase-shift algorithms have been developed.4,10,11 A popular five-step algorithm is based on five independent measurements of the intensity Ii corre-sponding to the five phase shifts ␾i=共i−3兲␲/ 2, where i = 1–5, as a result of which the calibration error is decreased: ⌬␸= tan−1关2共I2− I4兲/共2I3− I5− I1兲兴. 共2兲 The illuminating system has a substantial effect on the quality of the measurements. The advantages of interfero-metrically generated fringes are that they increase the illumi-nation efficiency, give a fringe contrast that is high and un-restricted in depth of the scene, and provide more flexible formation of the fringes. The fringes possess a pure sinu-soidal profile, and this provides high measurement accuracy. The effect of the external lighting under working conditions can be eliminated by using an appropriate interference filter. Liquid-crystal projectors with incoherent light and digital generation of fringes of various frequencies are characterized by high response rate but are limited in resolution. Plasma and micromirror devices create higher irradiance brightness. The best contrast is inherent to micromirror modulators, and this, along with high optical efficiency, makes them more promising for practical applications. The nonsinusoidal na-ture of the fringe profile because of the limited number and

size of the pixels is the main drawback of the indicated pro-jection systems. Moreover, defocusing in areas where the relief is of great depth reduces the fringe contrast and the measurement accuracy.

As follows from Eq. 共2兲, the phase differences are not determined unambiguously, but to within 2␲, and this re-quires joining the phase values at the break points caused by the tangential dependence. To correctly join them, an esti-mate of the quality of the phase data should be introduced in order to distinguish the break points and zones inherent to the object itself from those caused by noise, vibrations, shad-owing, etc. Good estimates of the quality of the individual pixels in the phase pattern can be obtained on the basis of the variance of the partial derivatives of the phase with respect to spatial coordinates x and y or the maximum value of the phase gradient, as well as by combining them.11 The maxi-mum phase gradient is determined in such a way that it ex-ceeded the maximum value of the partial derivatives of the phase for each pixel, where the maximum values are esti-mated by averaging within a k⫻k window for each pixel. In this paper, the joining of the phase is done using a method based on tracing the phase trajectory, for which, if there are no break points, the phase can be integrated along any trajectory.

MEASURING THE THREE-DIMENSIONAL COORDINATES BY THE METHOD OF PROJECTION OF TWO-PERIOD

INTERFERENCE PATTERNS

Three-dimensional coordinates can be efficiently mea-sured by projecting two-period interference patterns.12–14 This approach can be further improved by double symmetric illumination or double symmetric recording to reduce the influence of shading.

Figure 1 shows the layout for implementing a method of

projecting bright fringes by means of a Mach–Zehnder inter-ferometer 共Fig. 1a兲 and a micromirror projector 共Fig. 1b兲 with digital synthesis of the fringes. The method is based on the generation in the共x

⬘

, y

⬘

, 0兲 plane of interference fringes with periods d1and d2that are parallel to the Y

⬘

axis. The Y and Y

⬘

axes are perpendicular to the plane of the drawing. The phase of the projected fringes is determined as ␸i

⬘

= 2␲x

⬘

/ di, i = 1 , 2. The phase is reconstructed in the XYZ coordinate system, with the Z axis oriented parallel to the optical axis of the solid-state共CCD兲 recording camera. Angle

␣ is the slope angle of illumination axis Z

⬘

with respect to observation axis Z. Phase ⌬␸i共x,y兲 in the intensity distribu-tion given by Eq.共1兲 is determined by means of the five-step algorithm given in Eq. 共2兲 for each fringe pattern corre-sponding to a given period. The smaller period is chosen, allowing ten pixels per period, which in essence exceeds the theoretical requirement for accurately measuring the phase. The phase ⌬␸i共x,y兲 can be represented in the XYZ coordi-nate system as

⌬␸i共x,y兲 =␸i共x,y兲 −␸0=兵2␲关lx cos␣+ lz共x,y兲sin␣兴/di关l − z共x,y兲cos␣+ x sin␣兴其 −␸0, 共3兲 where i = 1 , 2; z共x,y兲 is the relief of the object at point 共x,y兲, l is the distance from the object to the exit pupil of the illumination objective, and ␸0 is an unknown calibration constant. Subtracting phase distributions ⌬␸i共x,y兲 and as-suming that

关␸2共x,y兲 −␸1共x,y兲兴/2␲⬅ Nx,y, 共4兲

where Nxyare numbers obtained on the basis of the measured phase values, we obtain the expression for coordinate z in the form

FIG. 1. Experimental apparatus for projecting fringes in coherent共a兲 and white 共b兲 light. LAS is a laser, O is the object, L is a lens, BS is a beamsplitter, SF is a spatial filter, P is a prism, MZ is a Mach-Zehnder interferometer, PD is a photodetector, PhD are photodiodes, PC is a computer, ADC is an analog-to-digital converter, CCD is a CCD camera, Pr is a micromirror projector, and DAC is a analog-to-digital-to-analog converter.

z共x,y兲 =

冋

Nxy共l + x sin␣兲 +d2− d1 d₁d₂ lx cos␣

册

冒

冋

Nxycos␣− d2− d1 d1d2 l sin␣

册

. 共5兲

The method’s sensitivity mainly depends on the accuracy with which the phase difference is measured, i.e., on the accuracy with which Nxyis determined. The influence of in-accuracy in determining l and␣ can be neglected.

Based on Eq. 共5兲, the sensitivity of the method, deter-mined by the minimum measurable difference of the z coor-dinates, can be written as

␦z =关d1d2共d1− d2兲/共x + l sin␣兲␦Nxy兴/关d1d2Nxycos␣

+共d₁− d₂兲/sin␣兲2_{. 共6兲}

The measurement accuracy increases with increasing differ-ence of the fringe periods d1− d2and with illumination angle

␣ but is not identical over the length of the object and de-creases as its transverse size inde-creases. Figure 2 shows the measurement indeterminacy 兩␦z兩 for a set of illumination angles ␣ as a function of period d2 at fixed values of d1 = 0.5 mm,␦Nxy= 0.02, and Nxy= 1.5.

The results of measuring the surface by the projection of two-period interference fringes 共Fig. 1a兲 at ␣= 70°, l = 1500 mm, d1= 0.5 mm, and d2= 2 mm are shown in Fig. 3. The object of measurement was a cylindrical vessel made from a composite material with an outer diameter of 200 mm under a pressure of 500 kP. The deformation of part of the side wall of the cylinder with dimension 100⫻100 mm was measured. Figure 3a contains segments of two-period fringes modulated by the surface of the object. The reconstructed phase difference in the intensity scale of white light modulo 2␲is shown in Fig. 3b. The corresponding joined image of the phase difference and a three-dimensional image are shown in Figs. 3c and 3d. The sensitivity of the method makes it possible to record the inhomogeneous deformation of the object’s surface under pressure.

Comparative experiments of the use of two-period pro-jection interferometry with generation of fringes by means of a Mach–Zehnder interferometer共Fig. 1a兲 and a micromirror

projector with digital synthesis 共Fig. 1b兲 were made on the same object—a 30⫻30⫻30-mm cube.15 In both experi-ments, the fringe periods were d1= 17.1 and d2= 8.5 mm,␣ = 30°, and l = 900 mm. The reconstructed phase difference and the corresponding joined images in a white-light inten-sity scale are shown in Figs. 4a and 4cwith generation of fringes in coherent light by a Mach–Zehnder interferometer and in Figs. 4b and 4d with digital generation of fringes by a micromirror projector in white light. It should be pointed out that the sensitivity and measurement accuracy in the given experiments are substantially worse than the results shown in Fig. 3, because of the smaller illumination angle ␣ and the impossibility of making the optimum choice of the periods of the interference fringes. This is associated with the limited resolution and the small size of the modulator of the micro-mirror projector 共800⫻600 pixels兲 for creating fringes with

FIG. 2. Measurement indeterminacy兩␦z兩 vs period d2.␣= 30°共1兲, 40° 共2兲,

50°共3兲, 60° 共4兲, 70° 共5兲. d1= 0.5 mm, x = 0,␦N = 0.02, l = 2000 mm, N = 1.5.

FIG. 3. Measurement of the absolute coordinates of the central part of a vessel made from a composite material using projection two-period interfer-ometry.共a兲 Segments of projected fringes with periods d1and d2,共b兲 phase

image modulo 2␲, 共c兲 reconstructed phase image, 共d兲 three-dimensional image of the surface.

a sinusoidal profile. A comparison of the results is shown in Fig. 5 for the points of intersection along line C–C, indicated in Figs. 4c and 4d. As expected, the profile-measurement accuracy by the method of interferometrically generated fringes is somewhat greater, but the method of projecting the fringes in white light is simpler and more convenient when solving a number of practical problems. To make measure-ments in real time by the fringe-projection method, it is nec-essary to illuminate the object simultaneously with phase-shifted fringes that are generated at different wavelengths, using interference beamsplitters during the recording.

THE HOLOGRAPHIC APPROACH TO PHASE RECONSTRUCTION

The method of interference-fringe projection presented here can be used to solve inverse problems in a dynamic holographic display, since the coordinates of the objects and scenes need to be measured in order to compute the diffrac-tion structures when reconstructing three-dimensional im-ages. The use of the method is limited mainly by the size of the objects and scenes, and this requires further improvement of the approaches when reconstructing the phase. Many phase-reconstruction methods have been proposed, based on

independent intensity measurements16,17of the wave front in the plane of the spatial coordinates共for example, in the im-age plane兲 and in the frequency plane. It is especially prom-ising to use the fractional Fourier transformation, as a gen-eralization of this approach to the case of multiple fractional Fourier transformations. Phase reconstruction only on the ba-sis of the Fourier transformation has also been investigated, with additional limitations with respect to size, nonnegativ-ity, etc. The possibilities of using multiple intensity measure-ments with a given coding 共modulation兲 of the input signal have been studied to a lesser degree. For this purpose, it is also possible to introduce or to use variations of the input signal introduced by the object. In fact, most phase-reconstruction methods are based on intensity measurements with two independent representations of the signal for com-parison of independent equations, which are solved itera-tively and usually require additional analysis.

In principle, with all phase-reconstruction methods, a light-intensity measurement is carried out for which informa-tion on the amplitude and phase of the wave front can be obtained, and this is characteristic of holography. However, the obvious connection of holography with phase-reconstruction methods has been indicated comparatively re-cently. A set of signals obtained by a given phase-reconstruction algorithm from measured intensities, with the introduction of additional limitations, can be taken in prac-tice for a complete description or “hologram” of the signal. The most general description of the signal is the image in-tensity, transformed or modified by some method. Thus, the recorded intensity and its Fourier representation or any of the fractional Fourier representations, taken together, constitute the hologram of the signal. Other sets of data can be consti-tuted by modulating, masking, or other suitable modification of the input signal. The intensities created by this modified input signal subsequently make it possible to reconstruct the phase and in this sense constitute a hologram of the signal. Based on these considerations, the solution of the problem of measuring the coordinates can reduce to comparing a set of modified images, as schematically shown in Fig. 6. In prin-ciple, there are many forms of modifications, including shift, slope, modulation, masking, and parametric transformation of the signal, for making independent measurements of the intensities and reconstructing the phase.

FIG. 4. Experimental results of the measurement of a test object by the method of fringe projection in coherent light共a and c兲 and in white light 共b and d兲. 共a and b兲 Phase image, 共c and d兲 reconstructed phase image.

FIG. 5. Comparison of cross sections indicated by line C–C in Figs. 4c and 4d. 1—Method of interferometrically generated fringes, 2—method of fringe projection.

This work was carried out under Contract 511568 “3DTV” at part of FP6-EC.

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